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and continuity practically where Eudoxus had left it in the .
fourth century B.C.; Kronecker, a modern echo of Zeno, made
Weierstrass' last years miserable by sceptical criticism of the
latter's revision of Eudoxus; while Cantor, striking out on a new
road of his own, sought to compass the actual infinite itself
which is implicit - according to some - in the very concept of
continuity* Out of the work of Weierstrass and Dedekind deve-
loped the modern epoch of analysis, that of critical logical
precision in analysis (the calculus, the theory of functions of a
complex variable, and the theory of functions of real variables)
in distinction to the looser intuitive methods of some of the
older writers - invaluable as heuristic guides to discovery but
quite worthless from the standpoint of the Pythagorean ideal
of mathematical proof. As has already been noted, Gauss, Abel,
and Cauchy inaugurated the first period of rigour; the move-
ment started by Weierstrass and Dedekind was on a higher
plane, suitable to the more exacting demands of analysis in the
second half of the century, for which the earlier precautions
were inadequate.
One discovery by Weierstrass in particular shocked the intui-
tive school of analysts into a decent regard for caution: he pro-
duced a continuous curve which has no tangent at any point.
Gauss once called mathematics *the science of the eye'; it takes
more than a good pair of eyes to 'see' the curve which Weier-
strass presented to the advocates of sensual intuition.
Since to every action there is an equal and opposite reaction
it was but natural that a]l this revamped rigour should engender
its own opposition. Kronecker attacked it vigorously, even
viciously, and quite exasperatingly. He denied that it meant
anything. Although he succeeded in hurting the venerable and
kindly Weierstrass, he made but little impression on his conser-
vative contemporaries and practically none on mathematical
analysis. Rronecker was a generation ahead of his tune. Not till
the second decade of the twentieth century did his strictures on
the currently accepted doctrines of continuity and irrational
numbers receive serious consideration. To-day it is true that
not all mathematicians regard Kronecker's attack as merely the
release of his pent-up envy of the more famous Weierstrass